$ F = \left[\begin{array}{rr}4 & 0 \\ -2 & 5\end{array}\right]$ $ E = \left[\begin{array}{rrr}0 & 0 & 2 \\ 1 & 3 & 4\end{array}\right]$ What is $ F E$ ?
Answer: Because $ F$ has dimensions $(2\times2)$ and $ E$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ F E = \left[\begin{array}{rr}{4} & {0} \\ {-2} & {5}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{0} & \color{#9D38BD}{2} \\ {1} & \color{#DF0030}{3} & \color{#9D38BD}{4}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{4}\cdot{0}+{0}\cdot{1} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{4}\cdot{0}+{0}\cdot{1} & ? & ? \\ {-2}\cdot{0}+{5}\cdot{1} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{4}\cdot{0}+{0}\cdot{1} & {4}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{3} & ? \\ {-2}\cdot{0}+{5}\cdot{1} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{4}\cdot{0}+{0}\cdot{1} & {4}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{3} & {4}\cdot\color{#9D38BD}{2}+{0}\cdot\color{#9D38BD}{4} \\ {-2}\cdot{0}+{5}\cdot{1} & {-2}\cdot\color{#DF0030}{0}+{5}\cdot\color{#DF0030}{3} & {-2}\cdot\color{#9D38BD}{2}+{5}\cdot\color{#9D38BD}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}0 & 0 & 8 \\ 5 & 15 & 16\end{array}\right] $